Optimal. Leaf size=500 \[ \frac{\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7}-\frac{\left (\frac{5 c}{a e}-\frac{9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6} \]
[Out]
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Rubi [A] time = 1.68289, antiderivative size = 500, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac{\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7}-\frac{\left (\frac{5 c}{a e}-\frac{9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**8/(e*x+d),x)
[Out]
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Mathematica [A] time = 1.23816, size = 557, normalized size = 1.11 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (105 x^7 \log (x) \left (c d^2-a e^2\right )^5 \left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right )-105 x^7 \left (c d^2-a e^2\right )^5 \left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (3 a^6 e^6 \left (5120 d^6+6400 d^5 e x+128 d^4 e^2 x^2-144 d^3 e^3 x^3+168 d^2 e^4 x^4-210 d e^5 x^5+315 e^6 x^6\right )+2 a^5 c d^2 e^5 x \left (18560 d^5+24320 d^4 e x+744 d^3 e^2 x^2-872 d^2 e^3 x^3+1099 d e^4 x^4-1680 e^5 x^5\right )+a^4 c^2 d^4 e^4 x^2 \left (23680 d^4+33520 d^3 e x+1824 d^2 e^2 x^2-2332 d e^3 x^3+3689 e^4 x^4\right )+60 a^3 c^3 d^6 e^3 x^3 \left (4 d^3+12 d^2 e x+5 d e^2 x^2-10 e^3 x^3\right )-35 a^2 c^4 d^8 e^2 x^4 \left (8 d^2+26 d e x+15 e^2 x^2\right )+350 a c^5 d^{10} e x^5 (d+4 e x)-525 c^6 d^{12} x^6\right )\right )}{215040 a^{9/2} d^{11/2} e^{9/2} x^7 (d+e x)^{3/2} (a e+c d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]
[Out]
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Maple [B] time = 0.107, size = 5353, normalized size = 10.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^8/(e*x+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^8),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^8),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**8/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 176.329, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^8),x, algorithm="giac")
[Out]